论文标题
$ k $的GCD连续斐波那契数
GCD of sums of $k$ consecutive squares of generalized Fibonacci numbers
论文作者
论文摘要
在2021年，盖尔（Guyer）和姆比里卡（Mbirika）给出了两个等效公式，该公式计算出$ k $的所有总和的最大分数（gcd），该$ k $连续术语中的fibonAcci序列$ \ left（g_n \ right）_ {n \ geq 0} $ exurrence $ g_ n $ g_ n = g_n = g_ g_ n = g_n = g_n = g_n = g_n = g_ g _ = \ geq 2 $具有整体初始条件$ g_0 $和$ g_1 $。在本文中，我们将其结果扩展到所有这些数字的$ k $连续广场的GCD。用符号$ \ MATHCAL {g} _ {g_1}^2 \！（k）$表示这些GCD值，我们证明$ \ Mathcal {g} _ {g_1，g_1}，g_1}^2 \！（k） g_ {k+1}^2 -g_1^2，\; g_ {k+2}^2 -g_2^2 \ right）$。此外，我们在斐波那契，卢卡斯和广义斐波那契数的特定设置中提供非常诱人的封闭形式。我们解决了许多开放问题，以供进一步研究。
In 2021, Guyer and Mbirika gave two equivalent formulas that computed the greatest common divisor (GCD) of all sums of $k$ consecutive terms in the generalized Fibonacci sequence $\left(G_n\right)_{n \geq 0}$ given by the recurrence $G_n = G_{n-1} + G_{n-2}$ for all $n \geq 2$ with integral initial conditions $G_0$ and $G_1$. In this current paper, we extend their results to the GCD of all sums of $k$ consecutive squares of these numbers. Denoting these GCD values by the symbol $\mathcal{G}_{G_0, G_1}^2\!(k)$, we prove $\mathcal{G}_{G_0, G_1}^2\!(k) = \gcd\left(G_k G_{k+1} - G_0 G_1,\; G_{k+1}^2 - G_1^2,\; G_{k+2}^2 - G_2^2\right)$. Moreover, we provide very tantalizing closed forms in the specific settings of the Fibonacci, Lucas, and generalized Fibonacci numbers. We close with a number of open questions for further research.
